What is 3/2 log_{4}(5t^6)-3/4 log_{4}(t^4) ?

The solution to 3/2 log_{4}(5t^6)-3/4 log_{4}(t^4) is log_{4}(5sqrt(5))+6log_{4}(t)

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simplify 3/2 log_{4}(5t^6)-3/4 log_{4}(t^4)

simplify

\frac{3}{2} lo g_{4} (5 t^{6}) - \frac{3}{4} lo g_{4} (t^{4})

lo g_{4} (55) + 6 lo g_{4} (t)

Solution steps

\frac{3}{2} lo g_{4} (5 t^{6}) - \frac{3}{4} lo g_{4} (t^{4})

Simplify

\frac{3}{4} lo g_{4} (t^{4}) : 3 lo g_{4} (t)

= \frac{3}{2} lo g_{4} (5 t^{6}) - 3 lo g_{4} (t)

\frac{3}{2} lo g_{4} (5 t^{6}) = lo g_{4} ((5 t^{6})^{\frac{3}{2}})

- 3 lo g_{4} (t) = - lo g_{4} (t^{3})

= lo g_{4} ((5 t^{6})^{\frac{3}{2}}) - lo g_{4} (t^{3})

Apply log rule:

lo g_{a} (x) - lo g_{a} (y) = lo g_{a} (\frac{x}{y})

lo g_{4} ((5 t^{6})^{\frac{3}{2}}) - lo g_{4} (t^{3}) = lo g_{4} \frac{( 5 t ^{6} ) ^{\frac{3}{2}}}{t ^{3}}

= lo g_{4} \frac{( 5 t ^{6} ) ^{\frac{3}{2}}}{t ^{3}}

Simplify

\frac{( 5 t ^{6} ) ^{\frac{3}{2}}}{t ^{3}} : 55 t^{6}

= lo g_{4} (55 t^{6})

Apply log rule:

lo g_{a} (x y) = lo g_{a} (x) + lo g_{a} (y)

lo g_{4} (55 t^{6}) = lo g_{4} (5) + lo g_{4} (5) + lo g_{4} (t^{6})

= lo g_{4} (5) + lo g_{4} (5) + lo g_{4} (t^{6})

Simplify

lo g_{4} (t^{6}) : 6 lo g_{4} (t)

= lo g_{4} (5) + lo g_{4} (5) + 6 lo g_{4} (t)

Apply log rule:

lo g_{a} (x) + lo g_{a} (y) = lo g_{a} (x y)

lo g_{4} (5) + lo g_{4} (5) = lo g_{4} (55)

= lo g_{4} (55) + 6 lo g_{4} (t)

- 9 lo g_{10} (y) + 7 lo g_{10} (x)

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What is 3/2 log_{4}(5t^6)-3/4 log_{4}(t^4) ?
The solution to 3/2 log_{4}(5t^6)-3/4 log_{4}(t^4) is log_{4}(5sqrt(5))+6log_{4}(t)